Question: Multiply the following complex numbers, marked as blue dots on the graph: $[\cos(\frac{4}{3}\pi) + i \sin(\frac{4}{3}\pi)] \cdot [4(\cos(\frac{1}{6}\pi) + i \sin(\frac{1}{6}\pi))]$ (Your current answer will be plotted in orange.)
Solution: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $\cos(\frac{4}{3}\pi) + i \sin(\frac{4}{3}\pi)$ ) has angle $\frac{4}{3}\pi$ and radius $1$ The second number ( $4(\cos(\frac{1}{6}\pi) + i \sin(\frac{1}{6}\pi))$ ) has angle $\frac{1}{6}\pi$ and radius $4$ The radius of the result will be $1 \cdot 4$ , which is $4$ The angle of the result is $\frac{4}{3}\pi + \frac{1}{6}\pi = \frac{3}{2}\pi$ The radius of the result is $4$ and the angle of the result is $\frac{3}{2}\pi$.